Due to that the effective Hamiltonian in Eq. (REF ) is a quadratic form, the evolution state will be a Gaussian state [1]} for the given initial vacuum state. Then the quantum Fisher information about the frequency \(\omega _1\)  can be calculated by [2]}
\(\mathcal {F}_{\omega _1}=\frac{1}{2}\frac{\textmd {Tr}[(C^{-1}C^{\prime })^2]}{1+P^2}+2\frac{P^{\prime 2}}{1-P^4}+(\partial _{\omega _1}\langle \vec{R}\rangle ^\mathcal {T})C^{-1}\partial _{\omega _1}\langle \vec{R}\rangle ,\) 
